11-5 Solving Two-Step Inequalities

Course 3

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Warm UpSolve.

1. 6x + 36 = 2x

2. 4x – 13 = 15 + 5x

3. 5(x – 3) = 2x + 3

4. + x =

x = –9

x = –28

x = 6

Course 3

11-5 Solving Two-Step Inequalities

78

316

1116

x = –

Problem of the Day

Find an integer x that makes the following two inequalities true:4 < x2 < 16 and x < 2.5

x = –3

Course 3

11-5 Solving Two-Step Inequalities

Learn to solve two-step inequalities and graph the solutions of an inequality on a number line.

Course 3

11-5 Solving Two-Step Inequalities

Solving a multistep inequality uses the same inverse operations as solving a multistep equation. Multiplying or dividing the inequality by a negative number reverses the inequality symbol.

Course 3

11-5 Solving Two-Step Inequalities

Solve and graph.

Additional Example 1A: Solving Two-Step Inequalities

4x + 1 > 13

4x + 1 > 13 – 1 – 1 Subtract 1 from both sides.

4x > 124x4

> 124

Divide both sides by 4.

x > 3 1 2 3 4 5 6 7

Course 3

11-5 Solving Two-Step Inequalities

Course 3

11-5 Solving Two-Step Inequalities

If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed.

Remember!

Additional Example 1B: Solving Two-Step Inequalities

–9x + 7 25

–9x + 7 25

– 7 – 7 Subtract 7 from both sides.

–9x 18

–9x–9

18–9

Divide each side by –9; change to .

x –2-6 -5 -4 -3 -2 -1 0

Course 3

11-5 Solving Two-Step Inequalities

Solve and graph.

Solve and graph.

Check It Out: Example 1A

5x + 2 > 12

5x + 2 > 12 – 2 – 2 Subtract 2 from both sides.

5x > 105x5

> 105

Divide both sides by 5.

x > 2 1 2 3 4 5 6 7

Course 3

11-5 Solving Two-Step Inequalities

–4x + 2 18

–4x + 2 18

– 2 – 2 Subtract 2 from both sides.

–4x 16

–4x–4

16–4

Divide each side by –4; change to .

x –4-6 -5 -4 -3 -2 -1 0

Check It Out: Example 1B

Course 3

11-5 Solving Two-Step Inequalities

Additional Example 2: Solving Inequalities That Contain Fractions

Multiply by LCD, 20.

8x + 15 18

– 15 – 15 Subtract 15 from both

sides.8x 3

Solve + and graph the solution.2x5

34

910

+ 2x5

34

910

20( + ) 20( )2x5

34

910

20( ) + 20( ) 20( )2x5

34

910

Course 3

11-5 Solving Two-Step Inequalities

Distributive Property.

Additional Example 2 Continued

x 38

8x8

38 Divide both sides by 8.

8x 3

0 1

38

Course 3

11-5 Solving Two-Step Inequalities

Check It Out: Example 2

Multiply by LCD, 20.

12x + 5 10

– 5 – 5 Subtract 5 from both

sides.12x 5

Solve + 3x5

14

510

+ 3x5

14

510

20( + ) 20( )3x5

14

510

20( ) + 20( ) 20 ( )3x5

14

510

Course 3

11-5 Solving Two-Step Inequalities

Distributive Property.

Check It Out: Example 2 Continued

x 512

12x12

512 Divide both sides by 12.

12x 5

0 5 12

Course 3

11-5 Solving Two-Step Inequalities

Additional Example 3: School Application

A school’s Spanish club is selling bumper stickers. They bought 100 bumper stickers for $55, and they have to give the company 15 cents for every sticker sold. If they plan to sell each bumper sticker for $1.25, how many do they have to sell to make a profit?

Let R represent the revenue and C represent the cost. In order for the Spanish club to make a profit, the revenue must be greater than the cost.

R > C

Course 3

11-5 Solving Two-Step Inequalities

Additional Example 3 Continued

The revenue from selling x bumper stickers at $1.25 each is 1.25x. The cost of selling x bumper stickers is the fixed cost plus the unit cost times the number of bumper stickers sold, or 55 + 0.15x. Substitute the expressions for R and C.

1.25x > 55 + 0.15x Let x represent the number of bumper stickers sold. Fixed cost is $55. Unit cost is 15 cents.

Course 3

11-5 Solving Two-Step Inequalities

Additional Example 3 Continued

– 0.15x – 0.15x Subtract 0.15x from both sides.

1.10x > 55

x > 50

The Spanish club must sell more than 50 bumper stickers to make a profit.

Divide both sides by 1.10.

1.25x > 55 + 0.15x

1.10x1.10

551.10>

Course 3

11-5 Solving Two-Step Inequalities

Check It Out: Example 3

R > C

A school’s French club is selling bumper stickers. They bought 200 bumper stickers for $45, and they have to give the company 25 cents for every sticker sold. If they plan to sell each bumper sticker for $2.50, how many do they have to sell to make a profit?

Let R represent the revenue and C represent the cost. In order for the French club to make a profit, the revenue must be greater than the cost.

Course 3

11-5 Solving Two-Step Inequalities

Check It Out: Example 3 Continued

The revenue from selling x bumper stickers at $2.50 each is 2.5x. The cost of selling x bumper stickers is the fixed cost plus the unit cost times the number of bumper stickers sold, or 45 + 0.25x. Substitute the expressions for R and C.

2.5x > 45 + 0.25x Let x represent the number of bumper stickers sold. Fixed cost is $45. Unit cost is 25 cents.

Course 3

11-5 Solving Two-Step Inequalities

– 0.25x – 0.25x Subtract 0.25x from both sides.

2.25x > 45

x > 20

The French club must sell more than 20 bumper stickers to make a profit.

Divide both sides by 2.25.

2.5x > 45 + 0.25x

2.25x2.25

452.25>

Check It Out: Example 3 Continued

Course 3

11-5 Solving Two-Step Inequalities

Lesson Quiz: Part I

Solve and graph.

1. 4x – 6 > 10

2. 7x + 9 < 3x – 15

3. w – 3w < 32

4. w +

x < –6

x > 4

Insert Lesson Title Here

w > –1623

14

12

w 38

1 2 3 4 5 6 7

-10 -9 -8 -7 -6 -5 -4

-18 -17 -16 -15 -14 -13 -12

0 38

Course 3

11-5 Solving Two-Step Inequalities

Lesson Quiz: Part II

5. Antonio has budgeted an average of $45 a month for entertainment. For the first five months of the year he has spent $48, $39, $60, $48, and $33. How much can Antonio spend in the sixth month without exceeding his average budget?

no more than $42

Course 3

11-5 Solving Two-Step Inequalities